($\hbar = 1.06 \times 10^{-34} \text{ J.s}$, mass of electron $\text{m}_e = 9.10 \times 10^{-31} \text{ kg}$, charge of electron $= 1.60 \times 10^{-19} \text{ C}$)
Copper Fermi Energy Calculation
The Fermi energy ($E_F$) for a free electron gas is determined by the electron number density ($n$), the reduced Planck constant ($\hbar$), and the electron mass ($m_e$). The formula used is:
$ E_F = \frac{\hbar^2}{2m_e} (3\pi^2 n)^{2/3} $
Fermi Energy Formula & Constants
Key parameters provided:
- Electron number density, $n = 8.3 \times 10^{28} \text{ m}^{-3}$
- Reduced Planck constant, $\hbar = 1.06 \times 10^{-34} \text{ J.s}$
- Mass of electron, $m_e = 9.10 \times 10^{-31} \text{ kg}$
- Charge of electron, $e = 1.60 \times 10^{-19} \text{ C}$ (for conversion to eV)
Step-by-Step Calculation
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Calculate the term $3\pi^2 n$:
First, compute the product $3\pi^2 n$ using the given electron density:
$ 3\pi^2 n = 3 \times \pi^2 \times (8.3 \times 10^{28}) $
$ 3\pi^2 n \approx 2.4575 \times 10^{30} \text{ m}^{-3} $
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Calculate $(3\pi^2 n)^{2/3}$:
Next, raise the result from Step 1 to the power of $2/3$:
$ (3\pi^2 n)^{2/3} = (2.4575 \times 10^{30})^{2/3} $
$ (3\pi^2 n)^{2/3} \approx 1.8156 \times 10^{20} \text{ m}^{-2} $
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Calculate the kinetic energy factor $\frac{\hbar^2}{2m_e}$:
Compute the term involving the constants $\hbar$ and $m_e$:
$ \frac{\hbar^2}{2m_e} = \frac{(1.06 \times 10^{-34} \text{ J.s})^2}{2 \times (9.10 \times 10^{-31} \text{ kg})} $
$ \frac{\hbar^2}{2m_e} \approx 0.6174 \times 10^{-38} \text{ J}^2\text{s}^2/\text{kg} $
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Calculate Fermi Energy in Joules ($E_F$):
Multiply the results from Step 2 and Step 3 to find the Fermi energy in Joules:
$ E_F = \left( \frac{\hbar^2}{2m_e} \right) \times (3\pi^2 n)^{2/3} $
$ E_F \approx (0.6174 \times 10^{-38} \text{ J}^2\text{s}^2/\text{kg}) \times (1.8156 \times 10^{20} \text{ m}^{-2}) $
$ E_F \approx 1.1203 \times 10^{-18} \text{ J} $
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Convert Joules to electron-volts (eV):
Use the conversion factor $1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}$ to express the energy in eV:
$ E_F (\text{eV}) = \frac{1.1203 \times 10^{-18} \text{ J}}{1.60 \times 10^{-19} \text{ J/eV}} $
$ E_F (\text{eV}) \approx 7.002 \text{ eV} $
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Rounding:
Rounding the result to one decimal place gives $7.0 \text{ eV}$.
The calculated Fermi energy of approximately $7.0 \text{ eV}$ falls within the provided correct answer range of 6.5 to 7.5 eV.
Important Questions from Free Electron Theory Fermi Energy Velocity
- Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is $E_F$. The energy corresponding to the filling of $\frac{N_A}{2}$ electrons, where $N_A$ is the Avogadro number, is $2^n E_F$. The value of $n$ is
Crystal structures of two metals A and B are two-dimensional square lattices with same lattice constant $a$. Electrons in metals behave as free electrons. The Fermi surfaces corresponding to A and B are shown by solid circles in figures.
The electron concentrations in A and B are $n_A$ and $n_B$, respectively. The value of $(\frac{n_B}{n_A})$ isIf $X$ is the dimensionality of a free electron gas, the energy ($E$) dependence of density of states is given by $E^{½X-Y}$, where $Y$ is ________.
- Potassium metal has electron concentration of $1.4 \times 10^{28}m^{-3}$ and the corresponding density of states at Fermi level is $6.2 \times 10^{46}$ Joule$^{-1} m^{-3}$. If the Pauli paramagnetic susceptibility of Potassium is $n \times 10^{-k}$ in standard scientific form, then the value of $k$ (an integer) is __________ (Magnetic moment of electron is $9.3 \times 10^{-24}$ Joule T$^{-1}$; permeability of free space is $4\pi \times 10^{-7}$ T m A$^{-1}$)
- Electronic specific heat of a solid at temperature $T$ is $C = \gamma T$, where $\gamma$ is a constant related to the thermal effective mass ($m_{eff}$) of the electrons. Then which of the following statements are correct?
